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Infinite Buffer Closed Lines - Practical Worst Case

To understand the practical worst case, it is important to understand the concept of a state.  We define a state to be a vector that indicates how many jobs are at each station.  For example, there are 20 possible states for a system with three jobs and four machines, as illustrated in the following table:
 
 
 

State  Vetor  State  Vector 
1  (3,0,0,0)  11  (1,0,2,0) 
2  (0,3,0,0)  12  (0,1,2,0) 
3  (0,0,3,0)  13  (0,0,2,1) 
4  (0,0,0,3)  14  (1,0,0,2) 
5  (2,1,0,0)  15  (0,1,0,2) 
6  (2,0,1,0)  16  (0,0,1,2) 
7  (2,0,0,1)  17  (1,1,1,0) 
8  (1,2,0,0)  18  (1,1,0,1) 
9  (0,2,1,0)  19  (1,0,1,1) 
10  (0,2,0,1)  20  (0,1,1,1) 

In the best case the system progresses through "spread out" states 17-20, while in the worst case the system progresses through "clumped up" states 1-4.  If we make the assumptions of
 

  1. a balanced line
  2. single machine stations
  3. exponential processing times
then all possible states become equally likely.  Hence under these conditions we can compute the flow time of the line by viewing the system from the perspective of an individual job.

Suppose there are N stations and w jobs in the (CONWIP) system, and the processing time at each station is t.  Then, when a "marked" job arrives at a station, the other w-1 jobs are evenly distributed among the N stations.  Therefore, on average, the marked job will see (w-1)/N jobs at the station and hence the expected time for the job to get through the station will be the sum of the process time for the marked job plus the process times for the jobs in queue, which is

t + [(w-1)/N]t

Since all N stations are identical, the total flow time for the line is computed by multiplying the above by N, which yields

FT = N[1+(w-1)/N]t = Nt + (w-1)t = T0 + (w-1)/rb

Note that we have made use of the fact that the raw process time is given by T0 = Nt and that the bottleneck rate is rb = 1/t.

By making use of Little's Law, TH = WIP/CT and the fact that W0 = rbT0, we can use the above to compute an expression for throughput as a function of WIP level as follows:

TH = [w/(w+W0-1)]rb

Example

We illustrate the behavior of the PWC using the four station line with identical parameters as before, that is:

Bottleneck rate rb = 0.5 parts/hour

Raw processing time T0 = 8 hours

Critical WIP W0 = 0.5 * 8 = 4 parts

Congestion co-efficient a = 1, for practical worst case,

Using the equations derived above, we can plot the flow time (FT) and throughput (TH) for the practical worst case and compare them to the best and worst cases as follows:.
 
Relationship between Flow Time and WIP for all the cases

 
Relationship between Throughput and WIP for all the cases

 

Since the PWC causes the system to progress through the good ("spread out") states of the best case, the bad ("clumped up") states of the worst case, and all states in between, we would expect the PWC to exhibit performance between that of the best and worst casess, which is what happens.  For a line with a given rb and T0, we define it to be "bad" if its performance is worse than the practical worst case and "good" if it is better than the practical worst case.

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